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A change of scale formula for Wiener integrals of cylinder functions on the abstract Wiener space. II. (English) Zbl 1160.28310

Summary: We show that for certain bounded cylinder functions of the form \[ F(x)=\widehat{\mu}((h_1,x)^\sim,\dots, (h_n,x)^\sim),\quad x\in B \] where \(\widehat{\mu}: \mathbb R^n\to\mathbb C\) is the Fourier-transform of the complex-valued Borel measure \(\mu\) on \(\mathcal B(\mathbb R^n)\), the Borel \(\sigma\)-algebra of \(\mathbb R^n\) with \(\|\mu\|<\infty\), the analytic Feynman integral of \(F\) exists, although the analytic Feynman integral, \[ \lim_{z\to -iq}I^{aw}(F;z)=\lim_{z\to -iq}(z/2\pi)^{n/2} \int_{\mathbb R^n}f(\vec u)\exp\{-(z/2)|\vec u| z^2\}\,d\vec u, \] do not always exist for bounded cylinder functions \(F(x)=f((h_1,x)^\sim,\dots,(h_n,x)^\sim)\), \(x\in B\). We prove a change of scale formula for Wiener integrals of \(F\) on the abstract Wiener space.
For part I, cf. Int. J. Math. Math. Sci. 21, No. 1, 73–78 (1998; Zbl 0891.28010).

MSC:

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46G12 Measures and integration on abstract linear spaces

Citations:

Zbl 0891.28010
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